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In this paper, we consider the following modified quasi-geostrophic equations$\partial_t\theta +\Lambda^\alpha\theta +u\vec\nabla\theta =0$, $u=\Lambda^{\alpha-1}\mathcal{R}^\perp(\theta)$ where $\alpha \in ]0,1[$ is a fixedparameter. This equation was recently introduced by P. Constantin, G. Iyer andJ. Wu in \cite{CIW} as a modification of the classical quasi-geostrophicequation. In this paper, we prove that for any initial data $\theta_\ast$ inthe Sobolev space $H^1(\mathbb{R}^2),$ the equation (MQG) has a global andsmooth solution $\theta $ in $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$

Global well-posedness for a Modified 2D dissipative quasi-geostrophic equation with initial data in the critical Sobolev space $H^1$